2.3.1
Polymeric fluids
Steady state and time-dependent shear banding in entangled polymer melts and solutions have been intensively studied in the last seven years since banding was first discovered to arise by Wang and co-workers in Ref. [168]. Actually, shear banding has been thought possible in entangled polymer melts for over two decades [155], following the prediction of a nonmonotonic constitutive curve for these materials in a model by Doi and Edwards [47]. This model considers a test chain to be constrained within a ‘tube’ of confinements provided by a mean field of entanglements with other polymers; an idea which formed the basis for many subsequent ‘tube’ based models [67, 78, 92, 105, 108, 109, 127].
In contrast, steady state shear banding in wormlike micellar solutions has been confirmed and studied experimentally for nearly two decades [16,70,71,76,89,95–97, 110,144], owing to a significantly smaller problem of edge fracture in these materials compared to entangled polymer melts (apparently due to their much smaller elastic moduli) [18, 151]. These results followed predictions of a nonmonotonic constitutive curve in a model by Cates [27, 155] that combined entangled polymer relaxation mechanisms from Doi and Edwards’ theory with the additional mechanism of ‘tube breaking’ specific to wormlike micelles, which can reversibly break and reform.
The most recent research on entangled polymer solutions or melts [17–20, 77, 136, 138, 161] and wormlike micelles [16, 76] shows that shear banding in startup is not limited to steady state flow, but also arises during the time-dependent response after the stress overshoot, i.e., during the negative slope of stress in strain ∂γΣ < 0. (This stress overshoot commonly arises in these materials for imposed shear rates that are faster than the rate at which stress is relaxed in the material.) Time-
dependent shear banding of this form was not only found to arise in samples that do shear band at steady state, where the magnitude of time-dependent shear banding is generally much greater than that found at steady state [18, 19, 136, 138, 162], but also found to occur in less well entangled polymer solutions that do not shear band at steady state [19, 77, 138]. It has also been shown that time-dependent shear banding can be so pronounced that elastic-like recoil with negative local velocities can arise [18, 19, 138].
Similar results of time-dependent shear banding associated with the stress over- shoot have been found in a two species elastic network model [174]; in molecular dynamics simulations of polymer melts [26]; and in the rolie-poly (RP) model [2–4] that can have either monotonic or nonmonotonic constitutive curves, depending on the value of its parameter β. The last of these is a model for entangled polymer melts based on the ‘tube’ theory of Doi and Edwards; we will fully describe the RP model in Chapter 3 for use throughout this thesis. A detailed investigation of time-dependent shear banding during shear startup in the rolie-poly model was performed by Adams et al. in Refs. [2–4]. The authors showed that time-dependent shear banding (including negative-velocity recoil) arose during the negative slope of shear stress in strain for shear startup at a rate in the region of the least slope of the constitutive curve, regardless of whether the curve was monotonic or not. They showed that this shear banding persisted to steady state for values of the parameters that gave nonmonotonic constitutive curves, and the magnitude of shear banding at steady state was weaker than it had been during the negative slope of shear stress in strain. The authors also showed that decreasing the entanglement number9 Z re- sulted in a smaller range of shear rates for which a nonmonotonic constitutive curve had a negative slope, so that for sufficiently small values of Z (at fixed β) the con- stitutive curve could be rendered monotonic. All of these results are (qualitatively) consistent with the experimental results described above.
In Chapter 5 we will derive a criterion for the onset of linear instability to shear heterogeneity for shear startup that is independent of material type, and which shows
9In experiment this corresponds to decreasing the level of entanglement of the polymer sample;
that the negative slope of shear stress in strain does indeed contribute towards linear instability (credit for this criterion is given to Dr. Suzanne Fielding). We will explore the use of this criterion in the RP and Giesekus models in order to explain their shear banding properties during shear startup. We will also show that the Giesekus model does not show the time-dependent shear banding properties described above for the RP model.
2.3.2
Soft glassy materials
In the Section 2.2.2 we outlined the generic rheological behaviour of soft glassy materials (SGMs), focussing on the response to a step stress deformation. We now build on that description of these materials with details of the response to the shear startup protocol.
We briefly noted above that thixotropic YSF display steady state shear banding for imposed shear rates below a critical value ˙γ < ˙γc[37,51, 107,114, 126, 141], which is thought to be due to an underlying negatively sloping branch of the constitutive curve that extends to the zero shear rate limit ˙γ → 0 [34–36, 57, 103, 106, 112, 128, 141]. However, it has also been shown that thixotropic YSF show time-dependent shear banding during the negative slope of shear stress in strain ∂γΣ < 0 after the overshoot, which returns to a state of homogeneity at steady state for imposed shear rates greater than this critical value ˙γ > ˙γc [107].
Transient shear banding that again arises during the negative slope of the shear stress in strain ∂γΣ < 0 has also been found in simple YSF that do not show steady state banding [43,45]. Similar results are found in numerics of a shear transformation zone model [100] and also in the SGR model [116], where both have monotonic constitutive curves. In the latter, the peak value of the shear stress at the overshoot (and also the strain at which it occurs) was shown to increase with the age of the sample10. This is consistent with experimental work on carbopol microgels that found the peak shear stress and strain to both increase logarithmically with the age of the sample and the imposed shear rate [43]. Similar results for the peak stress
10The age of the sample is the time elapsed between sample preparation and the onset of defor-
dependence on the waiting time and shear rate have also been found in molecular dynamics simulations of a Lennard-Jones glass [165].
In Chapter 5 we will investigate the response of a scalar fluidity model (that has a monotonic constitutive curve) to shear startup, and show that similar age- dependent transient shear bands to those found in experiments and simulations of the SGR model arise.
2.3.3
Glassy polymeric materials
Under extensional deformation at a constant strain rate ˙, polymer glasses show an overshoot of the tensile stress as a function of strain similar to that of the shear stress in strain of non-polymeric glasses in the shear geometry (described in Section 2.3.2). It has also been shown that the peak stress at the overshoot rises with the age of the sample and the imposed strain rate [84, 164], as found in the shear equivalent in non-polymeric glasses described in Section 2.3.2. However, in polymer glasses this tensile stress overshoot is followed by an indefinite rise of the stress due to strain hardening. The glassy polymer model shows qualitatively similar behaviour during extension at a constant strain rate ˙ [58]. Startup in the shear geometry is expected to give similar results of an overshoot and subsequent indefinite rise of the shear stress. We will show this to be true in the GP model during shear startup in Chapter 5, where we will also investigate the possibility of shear bands arising during the negative slope of shear stress in strain.